Dynamics of sparse Boolean networks with multi-node and self-interactions

We analyse the equilibrium behaviour and non-equilibrium dynamics of sparse Boolean networks with self-interactions that evolve according to synchronous Glauber dynamics. Equilibrium analysis is achieved via a novel application of the cavity method to the temperature-dependent pseudo-Hamiltonian that characterizes the equilibrium state of systems with parallel dynamics. Similarly, the non-equilibrium dynamics can be analysed by using the dynamical version of the cavity method. It is well known, however, that when self-interactions are present, direct application of the dynamical cavity method is cumbersome, due to the presence of strong memory effects, which prevent explicit analysis of the dynamics beyond a few time steps. To overcome this difficulty, we show that it is possible to map a system of N variables to an equivalent bipartite system of 2N variables, for which the dynamical cavity method can be used under the usual one time approximation scheme. This substantial technical advancement allows for the study of transient and long-time behaviour of systems with self-interactions. Finally, we study the dynamics of systems with multi-node interactions, recently used to model gene-regulatory networks (GRNs), by mapping this to a bipartite system of Boolean variables with two-body interactions. We show that when interactions have a degree of bidirectionality such systems are able to support a multiplicity of diverse attractors, an important requirement for a GRN to sustain multi-cellular life.